Appendix A: The Binomial Distribution

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1 730 Appendix A The Binomial Distribution Appendix A: The Binomial Distribution Suppose that we decide to record the gender of each of the next 25 newborn children at a particular hospital. What is the chance that at least 15 are female? What is the chance that between 10 and 15 are female? How many among the 25 can we expect to be female? These and other similar questions can be answered by studying the binomial probability distribution. This distribution arises when the experiment of interest is a binomial experiment that is, an experiment having the following characteristics. Properties of a Binomial Experiment 1. It consists of a fixed number of observations, called trials. 2. Each trial can result in one of only two mutually exclusive outcomes, labeled success (S) and failure (F). 3. Outcomes of different trials are independent. 4. The probability that a trial results in a success is the same for each trial. The binomial random variable x is defined as x number of successes observed when experiment is performed The probability distribution of x is called the binomial probability distribution. The term success here does not necessarily have any of its usual connotations. Which of the two possible outcomes is labeled success is determined by the random variable of interest. For example, if the variable counts the number of female births among the next 25 births at a particular hospital, a female birth would be labeled a success (because this is what the variable counts). This labeling is arbitrary: If male births had been counted instead, a male birth would have been labeled a success and a female birth a failure. For example, suppose that each of five randomly selected customers purchasing a hot tub at a certain store chooses either an electric model or a gas model. Assume that these customers make their choices independently of one another and that 40% of all customers select an electric model. Let s define the variable x number among the five customers who selected an electric hot tub This experiment is a binomial experiment with number of trials 5 P(S) P(E).4 where success (S) is defined as a customer who purchased an electric model. The binomial distribution tells us what probability is associated with each of the possible x values 0, 1, 2, 3, 4, and 5. There are 32 possible outcomes, and 5 of them yield x 1: SFFFF FSFFF FFSFF FFFSF FFFFS By independence, the first of these possible outcomes has probability P1SFFFF2 5 P1S2P1F2P1F2P1F2P1F

2 Appendix A The Binomial Distribution 731 The probability calculation is the same for any outcome with only one success (x 1). It does not matter where in the sequence the single success occurs. Thus p112 5 P1x P1SFFFF or FSFFF or FFSFF or FFFSF or FFFFS Similarly, there are 10 outcomes for which x 2, because there are 10 ways to select 2 outcomes from among the 5 trials to be the successes: SSFFF, SFSFF,..., and FFFSS. The probability of each results from multiplying together (.4) two times and (.6) three times. For example, and so P1SSFFF p122 5 P1x P1SSFFF2 1 c 1 P1FFFSS The general form of the distribution here is p1x2 5 P1x successes among the 5 trials2 number of outcomes probability of any particular 5 a ba with x successes outcome with x successes b 5 1number of outcomes with x successes21.42 x x This form was seen previously, where p(2) 10(.4) 2 (.6) 3. Let n denote the number of trials in the experiment. Then the number of outcomes with x successes is the number of ways of selecting x from among the n trials to be the success trials. A simple expression for this quantity is number of outcomes with x successes 5 n! x!1n 2 x2! where, for any positive whole number m, the symbol m! (read m factorial ) is de fined by m! 5 m1m 2 121m 2 22 p and 0! 5 1. The Binomial Distribution Let n number of independent trials in a binomial experiment p constant probability that any particular trial results in a success (continued)

3 732 Appendix A The Binomial Distribution Then p1x2 5 P1x successes among n trials2 n 5 x1n 2 x2 px 11 2 p2 n2x x 5 0, 1, 2, p, n The expressions a n x b or n! nc x are sometimes used in place of Both are x!1n 2 x2! read as n choose x, and they represent the number of ways of choosing x items from a set of n. The binomial probability function can then be written as p1x2 5 a n x bpx 11 2 p2 n2x x 5 0, 1, 2, p, n or p1x2 5 n C x p x 11 2 p2 n2x x 5 0, 1, 2, p, n Notice that the probability distribution is being specified by a formula that allows calculation of the various probabilities rather than by giving a table or a probability histogram. EXAMPLE A.1 Computer Sales Sixty percent of all computers sold by a large computer retailer are laptops and 40% are desktop models. The type of computer purchased by each of the next 12 customers will be noted. Define a random variable x by x number of computers among these 12 that are laptops Because x counts the number of laptops, we use S to denote the sale of a laptop. Then x is a binomial random variable with n 12 and p P(S).60. The probability distribution of x is given by 12! p1x2 5 x!112 2 x2! 1.62 x 1.42 n2x x 5 0, 1, 2, p, 12 The probability that exactly four computers are laptops is p142 5 P1x ! 4!8! If group after group of 12 purchases is examined, the long-run percentage of those with exactly 4 laptops will be 4.2%. According to this calculation, 495 of the possible outcomes (there are possible outcomes) have x 4. The probability that between four and seven (inclusive) computers are laptops is P14 # x # 72 5 P1x 5 4 or x 5 5 or x 5 6 or x 5 72

4 Appendix A The Binomial Distribution 733 Because these outcomes are disjoint, this is equal to P14 # x # 72 5 p142 1 p152 1 p162 1 p ! 4!8! c 1 12! 7!5! Notice that P14, x, 72 5 P1x 5 5 or x p152 1 p so the probability depends on whether or appears. (This is typical of discrete random variables.) The binomial distribution formula can be tedious to use unless n is very small. Appendix Table 9 gives binomial probabilities for selected n in combination with various values of p. This should help you practice using the binomial distribution without getting bogged down in arithmetic. Using Appendix Table 9 To find p(x) for any particular value of x: 1. Locate the part of the table corresponding to your value of n (5, 10, 15, 20, or 25). 2. Move down to the row labeled with your value of x. 3. Go across to the column headed by the specified value of p. The desired probability is at the intersection of the designated x row and p column. For example, when n 20 and p.8, p(15) P(x 15) (entry at intersection of x 15 row and p.8 column).175 Although p(x) is positive for every possible x value, many probabilities are 0 to three decimal places, so they appear as.000 in the table. There are much more extensive binomial tables available. Alternatively, most statistics computer packages and graphing calculators are programmed to calculate these probabilities. Sampling Without Replacement Suppose that a population consists of N individuals or objects, each one classified as a success or a failure. Usually, sampling is carried out without replacement; that is, once an element has been selected into the sample, it is not a candidate for future selection. If the sampling was accomplished by selecting an element from the population, observing whether it is a success or a failure, and then returning it to the population before the next selection is made, the variable x number of successes observed in the sample would fit all the requirements of a binomial random variable. When sampling is done without replacement, the trials (individual selections) are not independent. In this case, the number of successes observed in the sample does not have a binomial distribution but rather a different type of distribution called a hypergeo-

5 734 Appendix A The Binomial Distribution metric distribution. Not only does the name of this distribution sound forbidding, but also probability calculations for this distribution are even more tedious than for the binomial distribution. Fortunately, when the sample size n is much smaller than N, the population size, probabilities calculated using the binomial distribution and the hypergeometric distribution are very close in value. They are so close, in fact, that statisticians often ignore the difference and use the binomial probabilities in place of the hypergeometric probabilities. Most statis ticians recommend the following guideline for determining whether the binomial probability distribution is appropriate when sampling without replacement. Let x denote the number of successes in a sample of size n selected without replacement from a population consisting of N individuals or objects. If n # 0.05 (that is, if N at most 5% of the population is sampled), then the binomial distribution gives a good approximation to the probability distribution of x. EXAMPLE A.2 Security Systems In recent years, homeowners have become increasingly security conscious. A Los Angeles Times poll reported that almost 20% of Southern California homeowners questioned had installed a home security system. Suppose that exactly 20% of all such homeowners have a system. Consider a random sample of n 20 homeowners (much less than 5% of the population). Then x, the number of homeowners in the sample who have a security system, has (approximately) a binomial distribution with n 20 and p.20. The probability that five of those sampled have a system is p152 5 P1x entry in x 5 5 row and p 5.20 column in Appendix Table 9 1n The probability that at least 40% of those in the sample that is, 8 or more have a system is P1x $ 82 5 P1x 5 8, 9, 10, p, 19, or p182 1 p192 1 c 1 p c If, in fact, p.20, only about 3% of all samples of size 20 would result in at least 8 homeowners having a security system. Because P(x 8) is so small when p.20, if x 8 were actually observed, we would have to wonder whether the reported value of p.20 was correct. Although it is possible that we would observe x 8 when p.20 (this would happen about 3% of the time in the long run), it might also be the case that p is actually greater than.20. In Chapter 10, we showed how hypothesistesting methods could be used to decide which of two contradictory claims about a population (e.g., p.20 or p.20) is more plausible. The binomial formula or tables can be used to compute each of the 21 probabilities p(0), p(1),..., p(20). Figure A.1 shows the probability histogram for the binomial distribution with n 20 and p.20. Notice that the distribution is skewed to the right. (The binomial distribution is symmetric only when p.5.)

6 Appendix A The Binomial Distribution 735 p(x) FIGURE A.1 The binomial probability histogram when n 5 20 and p x Mean and Standard Deviation of a Binomial Random Variable A binomial random variable x based on n trials has possible values 0, 1, 2,..., n, so the mean value is m x 5 a 1x2p1x p p112 1 c 1 1n2p1n2 and the variance of x is s 2 x 5 a 1x 2m x 2 2 p1x m x 2 2 p m x 2 2 p112 1 c 1 1n 2m x 2 2 p1n2 These expressions appear to be tedious to evaluate for any particular values of n and p. Fortunately, algebraic manipulation results in considerable simplification, making summation unnecessary. The mean value and the standard deviation of a binomial random variable are m x 5 np and s x 5 "np11 2 p2 respectively. EXAMPLE A.3 Credit Cards Paid in Full It has been reported that one-third of all credit card users pay their bills in full each month. This figure is, of course, an average across different cards and issuers. Suppose that 30% of all individuals holding Visa cards issued by a certain bank pay in full each month. A random sample of n 25 cardholders is to be selected. The bank is interested in the variable x number in the sample who pay in full each month. Even though sampling is done without replace ment, the sample size n 25 is most likely

7 736 Appendix A The Binomial Distribution very small compared to the total number of credit card holders, so we can approximate the probability distribution of x by using a binomial distribution with n 25 and p.3. We have defined paid in full as a success because this is the outcome counted by the random variable x. The mean value of x is then m x 5 np and the standard deviation is s x 5 "np11 2 p2 5 " " The probability that x is farther than 1 standard deviation from its mean value is P1x,m x 2s x or x.m x 1s x 2 5 P1x, 5.21 or x P1x # 52 1 P1x $ p102 1 c 1 p152 1 p c 1 p using Appendix Table 92 The value of s x is 0 when p 0 or p 1. In these two cases, there is no uncertainty in x: We are sure to observe x 0 when p 0 and x n when p 1. It is also easily verified that p(1 p) is largest when p.5. Thus the binomial distribution spreads out the most when sampling from a population. The farther p is from.5, the less spread out and the more skewed the distribution. EXERCISES A.1 - A.16 A.1 Consider the following two binomial experi ments. a. In a binomial experiment consisting of six trials, how many outcomes have exactly one success, and what are these outcomes? b. In a binomial experiment consisting of 20 trials, how many outcomes have exactly 10 successes? exactly 15 successes? exactly 5 successes? A.2 Suppose that in a certain metropolitan area, 9 out of 10 households have cable television. Let x denote the number among four randomly selected households that have cable television, so x is a binomial random variable with n 4 and p.9. a. Calculate p(2) P(x 2), and interpret this probability. b. Calculate p(4), the probability that all four selected households have cable television. c. Determine P(x 3). A.3 The Los Angeles Times (December 13, 1992) reported that what airline passengers like to do most on long flights is rest or sleep; in a survey of 3697 passengers, almost 80% rested or slept. Suppose that for a particular route, the actual percentage is exactly 80%, and consider randomly selecting six passengers. Then x, the number among the selected six who rested or slept, is a binomial random variable with n 6 and p.8. a. Calculate p(4), and interpret this probability. b. Calculate p(6), the probability that all six selected passengers rested or slept. c. Determine P(x 4). A.4 Refer to Exercise A.3, and suppose that 10 rather than 6 passengers are selected (n 10, p.8), so that Appendix Table 9 can be used. a. What is p(8)? b. Calculate P(x 7). c. Calculate the probability that more than half of the selected passengers rested or slept. A.5 Twenty-five percent of the customers entering a grocery store between 5 p.m. and 7 p.m. use an express Bold excercises answered in back Data set available online Video Solution available

8 Appendix A The Binomial Distribution 737 checkout. Consider five randomly selected customers, and let x denote the number among the five who use the express checkout. a. What is p(2), that is, P(x 2)? b. What is P(x 1)? c. What is P(2 x)? (Hint: Make use of your computation in Part (b).) d. What is P(x 2)? A.6 A breeder of show dogs is interested in the number of female puppies in a litter. If a birth is equally likely to result in a male or a female puppy, give the probability distribution of the variable x number of female puppies in a litter of size 5. A.7 The article FBI Says Fewer Than 25 Failed Polygraph Test (San Luis Obispo Tribune, July 29, 2001) described the impact of a new program that requires top FBI officials to pass a polygraph test. The article states that false positives (tests in which an individual fails even though he or she is telling the truth) are relatively common and occur about 15% of the time. Suppose that such a test is given to 10 trustworthy individuals. a. What is the probability that all 10 pass? b. What is the probability that more than 2 fail, even though all are trustworthy? c. The article indicated that 500 FBI agents were tested. Consider the random variable x number of the 500 tested who fail. If all 500 agents tested are trustworthy, what are the mean and the standard deviation of x? d. The headline indicates that fewer than 25 of the 500 agents tested failed the test. Is this a surprising result if all 500 are trustworthy? Answer based on the values of the mean and standard deviation from Part (c). A.8 Industrial quality control programs often include inspection of incoming materials from suppliers. If parts are purchased in large lots, a typical plan might be to select 20 parts at random from a lot and inspect them. A lot might be judged acceptable if one or fewer defective parts are found among those inspected. Otherwise, the lot is rejected and returned to the supplier. Use Appendix Table 9 to find the probability of accepting lots that have each of the following (Hint: Identify success with a defective part): a. 5% defective parts b. 10% defective parts c. 20% defective parts A.9 An experiment was conducted to investigate whether a graphologist (handwriting analyst) could distinguish a normal person s handwriting from the handwriting of a psychotic. A well-known expert was given 10 files, each containing handwriting samples from a normal person and from a person diagnosed as psychotic. The graphologist was then asked to identify the psychotic s handwriting. The graphologist made correct identifications in 6 of the 10 trials (data taken from Statistics in the Real World, by R. J. Lar sen and D. F. Stroup [New York: Macmillan, 1976]). Does this evidence indicate that the graphologist has an ability to distinguish the handwriting of psychotics? (Hint: What is the probability of correctly guessing 6 or more times out of 10? Your answer should depend on whether this probability is relatively small or relatively large.) A.10 If the temperature in Florida falls below 32 F during certain periods of the year, there is a chance that the citrus crop will be damaged. Suppose that the probability is.1 that any given tree will show measurable damage when the temperature falls to 30 F. If the temperature does drop to 30 F, what is the mean number of trees showing damage in orchards of 2000 trees? What is the standard deviation of the number of trees that show damage? A.11 Thirty percent of all automobiles undergoing an emissions inspection at a certain inspection station fail the inspection. a. Among 15 randomly selected cars, what is the probability that at most 5 fail the inspection? b. Among 15 randomly selected cars, what is the probability that between 5 and 10 (inclusive) fail to pass inspection? c. Among 25 randomly selected cars, what is the mean value of the number that pass inspection, and what is the standard deviation of the number that pass inspection? d. What is the probability that among 25 randomly selected cars, the number that pass is within 1 standard deviation of the mean value? A.12 You are to take a multiple-choice exam consisting of 100 questions with 5 possible responses to each question. Suppose that you have not studied and so must guess (select 1 of the 5 answers in a completely random fashion) on each question. Let x represent the number of correct responses on the test. a. What kind of probability distribution does x have? Bold excercises answered in back Data set available online Video Solution available

9 738 Appendix A The Binomial Distribution b. What is your expected score on the exam? (Hint: Your expected score is the mean value of the x distribution.) c. Compute the variance and standard deviation of x. d. Based on your answers to Parts (b) and (c), is it likely that you would score over 50 on this exam? Explain the reasoning behind your answer. A.13 Suppose that 20% of the 10,000 signatures on a certain recall petition are invalid. Would the number of invalid signatures in a sample of size 1000 of these signatures have (approximately) a binomial distribution? Explain. A.14 A coin is to be spun 25 times. Let x the number of spins that result in heads (H). Consider the following rule for deciding whether or not the coin is fair: Judge the coin to be fair if 8 x 17 Judge the coin to be biased if either x 7 or x 18 a. What is the probability of judging the coin to be biased when it is actually fair? b. What is the probability of judging the coin to be fair when P(H).9, so that there is a substantial bias? Repeat for P(H).1. c. What is the probability of judging the coin to be fair when P(H).6? when P(H).4? Why are the probabilities so large compared to the probabilities in Part (b)? d. What happens to the error probabilities of Parts (a) and (b) if the decision rule is changed so that the coin is judged fair if 7 x 18 and judged unfair otherwise? Is this a better rule than the one first proposed? A.15 A city ordinance requires that a smoke detector be installed in all residential housing. There is concern that too many residences are still without detectors, so a costly inspection program is being contemplated. Let p the proportion of all residences that have a detector. A random sample of 25 residences will be selected. If the sample strongly suggests that p.80 (fewer than 80% have detectors), as opposed to p.80, the program will be implemented. Let x the number of residences among the 25 that have a detector, and consider the following decision rule: Reject the claim that p.8 and implement the program if x 15 a. What is the probability that the program is implemented when p.80? b. What is the probability that the program is not implemented if p.70? if p.60? c. How do the error probabilities of Parts (a) and (b) change if the value 15 in the decision rule is changed to 14? A.16 Exit polling has been a controversial practice in recent elections, because early release of the resulting information appears to affect whether or not those who have not yet voted will do so. Suppose that 90% of all registered California voters favor banning the release of information from exit polls in presidential elections until after the polls in California close. A random sample of 25 registered California voters will be selected. a. What is the probability that more than 20 will favor the ban? b. What is the probability that at least 20 will favor the ban? c. What are the mean value and standard deviation of the number of voters in the sample who favor the ban? d. If fewer than 20 in the sample favor the ban, is this at odds with the assertion that (at least) 90% of California registered voters favors the ban? (Hint: Consider P(x 20) when p.9.) Bold excercises answered in back Data set available online Video Solution available

10 Appendix B APPENDIX B Statistical Tables 740 Table 1 Random Numbers 740 Table 2 Standard Normal Probabilities (Cumulative z Curve Areas) 742 Table 3 t Critical Values 744 Table 4 Tail Areas for t Curves 745 Table 5 Curves of b 5 P(Type II Error) for t Tests 748 Table 6 Values That Capture Specifi ed Upper-Tail F Curve Areas 749 Table 7 Critical Values of q for the Studentized Range Distribution 753 Table 8 Upper-Tail Areas for Chi-Square Distributions 754 Table 9 Binomial Probabilities

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